Computing Hochschild cohomology of an algebra in GAP
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I would like to calculate Hochschild cohomology of a path algebra of a quiver (modulo some relations) using GAP.
Creating the quiver and its path algebra modulo relations is explained in detail in the QPA manual.
After having defined an algebra $A$ I used
M := AlgebraAsModuleOverEnvelopingAlgebra(A);
to regard $A$ as a module over its enveloping algebra $A otimes A^mathrmop$.
There are a few steps left to calculate Hochschild cohomology
- find a projective resolution $P^bullet$ of $M$
- take $operatornameHom (-, A)$ of the complex $P^bullet$ (or rather its truncation?)
- calculate the homology of the resulting complex
For step 1, there are both ProjectiveResolution and ProjectiveResolutionOfPathAlgebraModule. Which would be preferable here?
Step 2 I don't know how to do in GAP.
(Step 3 should be HomologyOfComplex(C,i); where C is the output from step 2.)
homology-cohomology gap quiver
asked Aug 14 at 10:16
Earthliŋ
743926
add a comment |Â
up vote
5
down vote
favorite
I would like to calculate Hochschild cohomology of a path algebra of a quiver (modulo some relations) using GAP.
Creating the quiver and its path algebra modulo relations is explained in detail in the QPA manual.
After having defined an algebra $A$ I used
M := AlgebraAsModuleOverEnvelopingAlgebra(A);
to regard $A$ as a module over its enveloping algebra $A otimes A^mathrmop$.
There are a few steps left to calculate Hochschild cohomology
- find a projective resolution $P^bullet$ of $M$
- take $operatornameHom (-, A)$ of the complex $P^bullet$ (or rather its truncation?)
- calculate the homology of the resulting complex
For step 1, there are both ProjectiveResolution and ProjectiveResolutionOfPathAlgebraModule. Which would be preferable here?
Step 2 I don't know how to do in GAP.
(Step 3 should be HomologyOfComplex(C,i); where C is the output from step 2.)
homology-cohomology gap quiver
asked Aug 14 at 10:16
Earthliŋ
743926
For a package-specific question, it might be better to ask QPA authors directly.
– Alexander Konovalov
2 days ago
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
I would like to calculate Hochschild cohomology of a path algebra of a quiver (modulo some relations) using GAP.
Creating the quiver and its path algebra modulo relations is explained in detail in the QPA manual.
After having defined an algebra $A$ I used
M := AlgebraAsModuleOverEnvelopingAlgebra(A);
to regard $A$ as a module over its enveloping algebra $A otimes A^mathrmop$.
There are a few steps left to calculate Hochschild cohomology
- find a projective resolution $P^bullet$ of $M$
- take $operatornameHom (-, A)$ of the complex $P^bullet$ (or rather its truncation?)
- calculate the homology of the resulting complex
For step 1, there are both ProjectiveResolution and ProjectiveResolutionOfPathAlgebraModule. Which would be preferable here?
Step 2 I don't know how to do in GAP.
(Step 3 should be HomologyOfComplex(C,i); where C is the output from step 2.)
homology-cohomology gap quiver
asked Aug 14 at 10:16
Earthliŋ
743926
I would like to calculate Hochschild cohomology of a path algebra of a quiver (modulo some relations) using GAP.
Creating the quiver and its path algebra modulo relations is explained in detail in the QPA manual.
After having defined an algebra $A$ I used
M := AlgebraAsModuleOverEnvelopingAlgebra(A);
to regard $A$ as a module over its enveloping algebra $A otimes A^mathrmop$.
There are a few steps left to calculate Hochschild cohomology
- find a projective resolution $P^bullet$ of $M$
- take $operatornameHom (-, A)$ of the complex $P^bullet$ (or rather its truncation?)
- calculate the homology of the resulting complex
For step 1, there are both ProjectiveResolution and ProjectiveResolutionOfPathAlgebraModule. Which would be preferable here?
Step 2 I don't know how to do in GAP.
(Step 3 should be HomologyOfComplex(C,i); where C is the output from step 2.)
homology-cohomology gap quiver
asked Aug 14 at 10:16
Earthliŋ
743926
edited 2 days ago
asked Aug 14 at 10:16
Earthliŋ
743926
asked Aug 14 at 10:16
Earthliŋ
743926
asked Aug 14 at 10:16
Earthliŋ
743926
743926
For a package-specific question, it might be better to ask QPA authors directly.
– Alexander Konovalov
2 days ago
add a comment |Â
For a package-specific question, it might be better to ask QPA authors directly.
– Alexander Konovalov
2 days ago
For a package-specific question, it might be better to ask QPA authors directly.
– Alexander Konovalov
2 days ago
For a package-specific question, it might be better to ask QPA authors directly.
– Alexander Konovalov
2 days ago
add a comment |Â
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For a package-specific question, it might be better to ask QPA authors directly.
– Alexander Konovalov
2 days ago